An ultra-bright atom laser

We present a novel, ultra-bright atom-laser and ultra-cold thermal atom beam. Using rf-radiation we strongly couple the magnetic hyperfine levels of 87Rb atoms in a magnetically trapped Bose-Einstein condensate. At low rf-frequencies gravity opens a small hole in the trapping potenital and a well collimated, extremely bright atom laser emerges from just below the condensate. As opposed to traditional atom lasers based on weak coupling, this technique allows us to outcouple atoms at an arbitrarily large rate. We demonstrate an increase in flux per atom in the BEC by a factor of sixteen compared to the brightest quasi-continuous atom laser. Furthermore, we produce by two orders of magnitude the coldest thermal atom beam to date (200 nK).


Introduction
Atom laser beams are coherent matter waves outcoupled from of a Bose-Einstein condensate (BEC). They are the matter-wave analogue to the photon laser, with the magnetic trap corresponding to the optical cavity and the freely propagating atoms to the laser beam. Soon after the demonstration of BEC in dilute gases [1,2], researchers demonstrated the first output couplers for matter-waves [3,4,5,6,7]. A series of beautiful experiments measured the spatial phase coherence [8] and the atom number correlations [9,10]. Matter waves have been coupled into the fundamental mode of matter-wave guides [11,12], and beam splitters [13] and Bragg reflectors [14] have been demonstrated. Recently, the arrival of a purely laser-cooled BEC has opened the possibility of a truly continuous atom-laser [15]. In addition to their fundamental interest there are promising applications for bright coherent atom lasers. They would allow higher sensitivity in matter-wave interferometry [16,17,18], which could be further enhanced by number squeezed atom laser beams [19,20,18]. The extremely tight focal spot of a coherent atom lasers could be exploited in direct atom lithography [21] or ultra-sensitive magnetometry with very high spatial resolution [22]. An extremely cold thermal atom beam would be very useful for high-resolution spectroscopy of ultra-cold collisions. A detailed review of the state-of-the-art can be found in [23].
A key factor in the performance of all of these devices is the spectral brightness and therefore the flux of the atom laser beam. In this respect the atom laser is superior to the thermal atom beam like the optical laser to the ordinary light bulb. The flux of the traditional atom-laser achievable from a given condensate has been fundamentally limited by the outcoupling process. In this paper we present a novel output-coupler for magnetically trapped atoms, which eliminates this limit. We demonstrate an increase in the flux of the atom laser by a factor of seven compared to the brightest atom lasers demonstrated so far. This novel output coupler allowed us also to produce a thermal beam, which is by two orders of magnitude the coldest every observed.

Output coupling using weak fields
Atom laser beams can be generated in a number of different ways: Pulsed atom lasers have been produced from magnetically trapped BECs by short blasts of rf-radiation † [3]. An atom laser based on spilling atoms from a purely optical potential has been demonstrated for small laser fluxes [7]. Bright quasi-continuous atom lasers are usually outcoupled from magnetically trapped BECs using weak electromagnetic fields. This couples the Zeeman levels of the atoms and thus the trapped states to propagating ones. The transition can either be direct (rf output coupler) [6] or via a virtual state (Raman † The rf in [3] was pulsed in intensity or swept in frequency at a rate of 500 MHz/s, which means that in both cases the atoms released on a time scale much shorter than a quarter oscillation of the atoms in the magnetic trap. output coupler) [5]. Even though this process is irreversible [24], it is coherent in the sense that the atoms retain a well determined phase relationship with the other atoms in the beam and in the condensate [25]. After the atoms have been transferred to the propagating states they accelerate out of the condensate and form the atom laser beam. This acceleration is due to a combination of gravity, magnetic forces, and interaction with the atoms remaining in the condensate. For very weak fields, the flux of the atom laser is proportional to the intensity of the rf-field [26]. For stronger coupling fields, however, a bound state appears, which eventually shuts off the lasers thus severely limiting the maximal atom flux that can be achieved from a given condensate [27,28].
In the weakly coupled atom laser, the outcoupling occurs wherever the rf frequency is resonant with the local magnetic field. If this happens somewhere inside the condensate, then the chemical potential of the remaining condensate will act as as a matter-wave lens, which can lead to sever distortions of the atom-laser beam [29].

Output coupling using strong fields
Here we present a novel type of continuous atom laser, which is based on time-dependent adiabatic potentials (TDAP). Rather than using a weak rf-field to slowly outcouple a small fraction of the condensate, we use strong rf-fields to deform the trapping potential such that the condensate spills slowly out of a small opening at the bottom of the trap. As opposed to the atom laser based on the weak-field output coupler, the TDAP can emit atoms from the condensate at an almost arbitrary rate. This allowed us to generate atom-laser beams, which have a much larger flux than can be achieved in the weak coupling regime. The TDAP also permitted us to produce the coldest atom beam observed so to date. Examples can be seen in Fig. 1: The central panel shows an extremely bright atom laser, the left panel a highly collimated atom laser, and the right image for the first time a combined thermal and atom-laser beam.

Theory
The adiabatic potentials of the TDAP atom laser are created by subjecting magnetically trapped atoms (here 87 Rb, F=2, m F =-2) to a strong rf-field. Atoms, which pass through the region where they are resonant with the rf-field, are adiabatically transferred from their trapped to their anti-trapped state, thus limiting the trap depth ( Fig. 2). At higher rf-frequencies this is used in forced evaporative cooling, where one removes the hottest atoms of a trapped cloud by slowly ramping down the rf-frequency. At lower rf-frequencies the gravitational tilt of the potential causes the atoms to escape preferentially downwards producing to a thermal atom beam. As the trap depth approaches the chemical potential of the BEC, an atom laser beam emerges from the bottom edge of the condensate. Since the atoms are adiabatically transferred from the trapped to the anti-trapped state, all atoms of the BEC enter the atom laser beam. The flux of the TDAP atom laser is therefore simply a function of the rate at which the rf-frequency is ramped down ‡.
We focus our description of TDAP atom lasers on cigar-shaped Ioffe-Pritchard type magnetic traps. If the radial trapping frequency (ω ρ ) is much larger than the axial one (ω z ), then the magnetic field of a Ioffe-Pritchard trap can be written as B (r) = α x, −α y, B 0 + 1 2 βz 2 , where α is the gradient of the radial quadrupole field and β is the curvature of the axial field. The Larmor frequency associated with the difference in energy between adjacent Zeeman levels is Ω L = |g F µ B B(r)| /h, where g F is the Landé g-factor of the considered hyperfine manifold and µ B is the Bohr magneton. We couple the magnetic hyperfine states using a linearly polarised oscillating magnetic field of strength B rf and angular dressing frequency ω rf . The resulting rf coupling strength is Ω rf = |g F µ B /2h| · | B(r)/|B(r)| × B rf |.
In the weakly coupled atom laser the atomic motion quickly renders irreversible the transfer of the atoms to other spin states [24]. In the TDAP atom laser, however, the coupling strength is large compared to the change in Larmor frequency due to the motion of the atoms (Ω 2 rf Ω L /2π). Therefore, the spin-transfer is fully adiabatic.
We can then ignore the external degrees of freedom and use dressed states to describe ‡ Note that as opposed to the TDAP, the weakly coupled regime transfers the atoms irreversibly into a number of different magnetic hyperfine states [24]. the atom in the fields [30,31]. Taking into account gravity we can write the adiabatic potential as [30,32,31]: where M is the mass of an atom, g e earth's gravitational acceleration, and m F the magnetic quantum number of the total atomic spin when Ω L is large (Ω L ω rf , Ω rf ). Fig. 2 shows the dressed Zeeman states of Eq. (1) for a spin two particle in a harmonic magnetic trap. The left panel shows the potential in the horizontal (x or z) direction and the right panel in the vertical (y) direction. The arrows indicate the position where the rf is resonant with the local magnetic field. The maximum energy of a trapped condensate (the minimum energy, at which the atoms escape into the atom laser beam) is indicated by the dotted horizontal line. Fig. 3 shows a contour plot of the trapping potentials relative to the trap bottom § for the m F = −2 state for three different values of ω rf : a) where the rf-frequency is well above the trap bottom and the trap is relatively deep, b) where the rf-frequency is smaller and the trap more shallow, and c) where the rf-frequency is such that the trap just ceases to exist.
The trap depth can be adjusted dynamically simply by changing the value of ω rf : As one lowers the rf frequency, atoms which had been trapped at an energy larger than the new trap depth escape at the bottom of the trap. The rate at which one can change ω rf -and thus the rate at which atoms are outcoupled -is only limited by the requirement that the spin flips be adiabatic. This limits the ramping rate of the rf-frequency toω rf Ω 2 rf . For our typical coupling strength of Ω rf /2π = 16 kHz this imposesω rf /2π 1.6 × 10 9 Hz/s. A condensate with a typical chemical potential of the order of kHz could therefore be released within 1 µs, which is much faster than the § The rf-frequency relative to the trap bottom is the rf-frequency of the dressing field minus an offset, which is chosen such that at zero trap depth the rf-frequency relative to the trap bottom is zero. The trap parameters are the ones of Fig. 1a. The three plots have decreasing rffrequencies (∆ω rf /2π) with a) being the deepest and in c)just about ceasing to exist. In a) and b), the rf-frequency is 30 kHz and 15 kHz above the one of c). The equipotential lines are spaced by 100 nK and the colour scale is relative to the trap bottom. Note that the potential in the y-z plane is very similar to the one displayed here, except for a much elongated z-axes.
radial oscillation time. This stands in stark contrast to the weakly coupled atom laser, where the outcoupling rate is limited by the appearance of bound states [24].

Shape of the atom laser beam
The exact shape of the outcoupled atom beam is difficult to predict analytically. However, we can distinguish different regimes of output coupling by comparing the trap frequencies (ω z , ω ρ ) to the output coupling rate of the condensate (Ω oc ), which represents the inverse of the time that it takes to outcouple the entire condensate. At output coupling rates well below the axial trapping frequency (Ω oc ω z ω ρ ), the shape of the condensate can adapt adiabatically to the reducing atom number [33]. The output coupling will then only occur in a very small region below the centre of the condensate. At larger outcoupling rates (Ω oc ω z ω ρ ) the outcoupling region grows larger and shape oscillations are excited in the condensate, which affects the intensity and the shape of the atom laser beam. When the output coupling rate is large compared to the axial frequency but is still small compare to the radial trapping frequency (ω z Ω oc ω ρ ), then the shape in the axial direction remains essentially frozen [21,34] and the output coupling will occur below the whole length of the cigarshaped condensate. Finally, when the sweep of the rf-frequency is fast compared to the radial trapping frequency (ω z ω ρ Ω oc ) then the atoms will form a single accelerating shell-shaped matter-wave pulse [3].
A number of factors lead to the atom laser being very well collimated both in the slow (Ω oc ω z ω ρ ) and intermediate (ω z Ω oc ω ρ ) outcoupling regimes: The absence of matter-wave lensing, the smooth shape of the potential, the absence of shape oscillations, and the vertical acceleration of the atoms after they leave the trap: Matter wave lensing occurs in atom lasers based on a weak rf-field. When the laser beam is outcoupled closer to the centre of the condensate the chemical potential of the remaining condensate acts as a matter-wave lens, which distorts the atom laser beam. For the TDAP atom laser, however, matter wave lensing is completely absent because the atoms are outcoupled from the very edge of the condensate where the chemical potential is negligible. Furthermore, the shape of the confining potential at the outcoupling point tends to be very smooth in the direction transverse to the atom beam and the transverse gradient and curvature of the adiabatic potential become vanishingly small for very low output coupling rates (Ω oc ω z ω ρ ). Shape oscillations of the condensates are absent because in slow and intermediate outcoupling regimes the internal dynamics of the condensate are negligible. Finally, since the outcoupled atoms are in the anti-trapped magnetic hyperfine state they are strongly accelerated downward, thus minimising the influence of the transverse momentum component.
Numerical simulations of the time-dependent Schrödinger equation confirm that the TDAP atom laser can produce well-collimated quasi-continuous atom-laser beams (see appendix 7.4).

Longitudinal velocity
For many applications it is desirable for the atom laser beam to have a low transverse and a large longitudinal velocity. Examples include collision physics [35,36], surface science [37,38] and atom nano-lithography [39]. The acceleration of the TDAP atom laser due to the atoms being in the anti-trapped state (m F = −2) provides these high velocities, reaching 2 m s −1 after only 1 cm of travel.
For other applications a slower beam might be more useful. In this case it is sufficient to introduce a second rf-field, which transfers the atoms from the m F = −2 back to the m F = +2 state, which causes the atom laser beam to decelerate and eventually come to a standstill thus providing access to a very large range of longitudinal velocities [40].

Image analysis
Absorption images of the atom laser in the y-z plane are taken by shining resonant laser light along the x-axes, image it onto a CCD camera, and calculate the atom column density for each pixel. We then cut the images into horizontal slices, integrate each slice in the vertical direction, and fit them with the following binomial profile: The first term accounts for any global offset. The second term represents the Gaussian momentum distribution of a thermal contribution. The third term has been chosen ad-hoc for the apparent absence of wings in some of some of the experimental images of the atom laser. It implies an inverted-parabola shape of the transverse density profile , which is integrated in the direction of the imaging beam. Examples of such fits can be found in the supplementary material.
From the fits to the individual integrated atom slices we can then determine the atom number and sizes of both the coherent and thermal components. From the position of a slice we can calculate the velocity and output coupling time of the atoms in it and subsequently the divergence, temperature, and local flux of the atom beam. If no reliable fit can be obtained for the bimodal fit, we force either b = 0 or c = 0 in order to obtain a fit for only an atom-laser or only a thermal beam respectively.

Making the atom-laser and BEC
The creation of the TDAP atom laser is rather straight forward: A linear sweep of the rf-frequency cools a thermal cloud of magnetically trapped atoms into a BEC and then outcouples the atom laser from the trap.
Our experimental apparatus has been described elsewhere [41,42]. We load 10 9 atoms into a shallow magnetic Ioffe Pritchard trap. Within one second we compress this trap to its final state by ramping up the currents in the Ioffe and pinch coils whilst simultaneously reducing B 0 to 0.5-1 G. The final trapping frequencies are then 17 Hz in the axial and 561-793 Hz in the radial direction. Then we lower the rf-frequency down to its final value in a linear ramp of 10 s duration (Fig. 3a-c). The sample cools by forced evaporation until the critical temperature is reached and a BEC of about 1.5×10 5 atoms forms. A thermal atom beam emerges from the bottom of the trap. As the rffrequency drops even lower, the trap depth becomes smaller than the chemical potential of the condensate and an atom-laser is outcoupled from the lower edge of the condensate (Fig 3b) until finally the trap vanishes (Fig. 3c). We then switch off the magnetic trap, let the atoms evolve in free flight for 1-10 ms, and take a resonant absorption picture of y-z plane.
Note that the requirements on the reproducibility of B 0 are relatively low: Whereas the traditional atom laser needs to aim the weak rf exactly to the bottom of the BEC, the TDAP atom-laser only needs to sweep across it. In our case the magnetic field reproducibility is of the order of a few milli-Gauss with a shot-to-shot noise of less than one milli-Guass and a 50 Hz modulation of 40 mG.

Atom Lasers
In this section we describe in detail three different atom lasers: A well-collimated pure atom laser (Fig. 1a), an atom laser with a very high atom-flux (Fig. 1b), and finally an atom beam containing at the same time an atom laser and a thermal atom beam (Fig. 1c). The trap had a gradient α = 440 G/cm and a curvature β = 170 G/cm 2 . The rf-frequency was ramped down from 50 MHz at a speed ofω rf /2π = 5 MHz/s. The output coupling was therefore at an intermediate rate (ω z Ω oc < ω ρ ). We generate the rf with a direct digital synthesis card (Analog Devices AD9854/PCBZ), amplify it  using an rf amplifier (Amplifier Research 25A250A), which is coupled to two antennas in parallel (4 cm diameter, three windings each).
he three experiments described below differ in the initial temperature, the atom number, the offset field (B 0 ), and the duration of the time-of-flight free expansion (1-10 ms).

A well-collimated pure atom laser:
A well-collimated pure atom laser of ∼ 4.5 mm length and 2 ms duration can be seen in Fig. 1a. It originated from a trap with non-dressed axial and radial trapping frequencies of 16.6 Hz and 561 Hz, respectively, and is therefore in the regime ω z Ω oc ω ρ . Since there are no observable thermal wings to the atom laser we fit only the atom-laser part of Eq. (2) and find a peak flux of 2.5 × 10 7 atoms per second for a total of 3.1 × 10 4 atoms.
As discussed in section 7.3, we expect the TDAP to be very well collimated. Fitting a straight line to the width as a function of distance (Fig. 4), we find a divergence of only 10 mrad, which compares well with the lowest divergences reported for any atom laser [43,44]. The width of the atom laser at its origin is 50±5 µm. A condensate containing all 3.1 × 10 4 atoms detected in the atom-laser beam would have an axial Thomas-Fermi radius of 44 µm. The agreement between the width of the atom laser beam and the Thomas-Fermi radius in the trap confirms that in the regime ω z Ω oc ω ρ the condensate dynamics remain frozen in the axial direction and the outcoupling occurs   (Fig.1b). The horizontal dotted line shows the highest flux achieved previously [24]. In b) at a position of 1.5 mm below the condensate we observe a step-change of the transverse size of the fit, which we attribute to the onset of the atom laser emission. The dashed vertical line serves as a guide to the eye. A linear fit to the atom-laser beam sizes (full line in b) results in a divergence of Θ = 22 mrad. The size at y = 0 is 62 µm, which is very close to the Thomas-Fermi radius of 66 µm for 1.4 × 10 5 atoms in the non-dressed trap (16.6 Hz × 793 Hz).
over the full width of the condensate.

A very high flux atom laser:
Much effort has been focused on trying to maximise the flux available from a given BEC [45,46,24]. The main obstacle to achieving very large fluxes from magnetically trapped BECs is the appearance at large coupling strengths (Ω 2 rf →Ω L /2π) of a bound state, which shuts off the atom laser [27,28]. The reason for this is that it is not possible by ramping up the rf-intensity to adiabatically transfer the atomic population of a bare state to a single dressed state. In contrast to this, the TDAP atom laser relies on a strong rf, where the rf-frequency is scanned. This transfers the entire population adiabatically from the trapped to the anti-trapped state. We verified this in a Stern Gerlach experiment, where we found all atoms in the m F = −2 state after the rf-frequency had swept through the bottom of the trap. We also confirmed by rf-spectroscopy that the chemical potential of the original condensate matches the atom number detected in the atom laser beam.
The flux and transverse size of the atom laser of Fig. 1b can be seen in Fig. 5a) and b). The horizontal axis is the position below the condensate after a free expansion of 1 ms. Atoms imaged at higher values of the position have been outcoupled at higher dressing frequencies (ω rf ). Since there are no discernible thermal wings, we fit the atomlaser part of Eq. (2) only. For smaller values of ω rf , i.e. smaller distances in Fig. 5b, we find a well collimated beam. At about 1.6 mm, however, we find a sudden increase in the divergence of the beam. The signal to noise ratio below this point did not permit us to analyse the exact shape of the atom beam. The absence of thermal wings for lower rf-frequencies together with the sudden change in divergence of the atom beam indicate an onset of atom-lasing, i.e. the instant when the trap depth becomes smaller than the chemical potential of the BEC.
The flux in Fig. 5a reaches 7.4 × 10 7 atoms/s originating from only 9 × 10 4 atoms in the BEC. This is more than seven times larger than the previous maximum flux even with our initial BEC having only less than half as many atoms [24].

4.2.
3. An ultra-cold thermal beam: Fig. 1c shows for the first time an atom beam, which contains concurrently an atom laser and a thermal atom beam. The upper part of Fig. 1c is an atom laser beam whereas the lower part is an ultra-cold thermal beam. In the central part of the figure the thermal and the atom laser overlap. We analyse the image by fitting the full Eq. (2) to the integrated image slices. A plot of the integrated slices and fits can be found in the supplementary material. We find 7.2 × 10 4 atoms in thermal beam and 2.1 × 10 4 atoms in the atom laser. Fig. 6a plots the thermal and atom laser fluxes against the position of the slice (bottom axis) and the value of the rf-frequency at the time of output coupling (top axis). As the rf-frequency ramps down (right to left on the plot), initially only thermal atoms are outcoupled with a peak rate of 1.5 × 10 8 atoms/s. At an rf-frequency of about 5 kHz above the trap bottom atom laser emission sets in. It reaches a flux of up to 9 × 10 7 atoms/s until the trap opens up completely. The divergence of the atom laser is 22 mrad and its duration 0.35 ms. Fig. 6b shows the temperature of the thermal beam as calculated from its width and expansion time. The solid line is the critical temperature of Bose-Einstein Condensation (450 nK) in the non-dressed trap for the total number of atoms both beams. The temperature of the thermal beam stays constant because at this point the collision rate is no longer sufficient to thermalise the atoms remaining in the trap. At only 200 nK this is-to our best knowledge-the coldest thermal atom beam reported to date by more than two orders of magnitude [47,48,49,50]. Even though the absolute flux of the thermal atoms is very low compared to other atom beam sources, the extremely high energy resolution possible with this ultra-cold thermal beam will be useful e.g. in experiments comparing scattering of coherent and thermal atoms.

Conclusions
We demonstrated a novel atom laser based on time-dependent adiabatic potentials. It allows an output coupling of the atoms at almost arbitrarily large rates and thus eliminates the bottleneck of the bound states in the traditional output couplers based on weak coupling. Our maximum atom laser flux of 7.4 × 10 7 atoms per second exceeds the maximum demonstrated anywhere by a factor of seven. Larger condensates and faster rf ramp rates will allow us to push this in the near future by another order of magnitude.
We observed for the first time an atom beam containing both a thermal and an atom-laser component. Its temperature of only 200 nK is by more than two orders of magnitude the coldest thermal beam produced by any other technique.

Acknowledgments
We acknowledge the financial support of the

Flux Calculation
Near the trap bottom (y B 0 ) we can write the trapping potential in the direction of gravity (x = 0, z = 0) as where ∆ω rf = ω rf −|g F | µ B B 0 /h is the detuning of the rf-frequency from the trap bottom in the absence of the dressing field, and α ω is the gradient of the radial quadrupole field written in angular frequency units (α ω = g F µ B α/h). If the rf-field is linearly polarised and orthogonal to the z-axis, then close to the centre of an elongated Ioffe-Pritchard trap the coupling strength can be simplified to Ω rf = |g F | µ B B rf /(2h). We calculate the trap depth (∆V trap ) by taking the difference (∆V trap = V out − V 0 ) of the trap minimum (V 0 ) and output coupling point (V out ). Fig. 7 shows the trap depth for our typical experimental parameters as a function of the coupling strength Ω rf and of the rf-frequency relative to the trap bottom. Note that because of gravity and the weakening of the trap by the dressing field, the depth of shallow traps is much smaller than the detuning of the rf-frequency from B 0 (∆V trap h ∆ω rf ). During the lasing phase the atoms are accelerated both by gravity (g e ) and by the gradient of the magnetic field (a α = m F g F µ B α/M ). During the time-of-flight expansion they feel only earth's acceleration. The atoms in the slice located at position x of the image have been accelerated by the magnetic gradient for a time: For convenience, we set x=0 at the position of the BEC after time-of-flight expansion of duration (t E ) . The velocity of the atoms just after the time-of-flight expansion is given by v E = (g e + a α ) t L + g e t E . The flux just after the lasing phase but before the time-of-flight expansion is then where n 1D is the one dimensional density of the atoms, i.e. the number of atoms conatained in a slice divided by its thickenss.

Temperatures
The temperature of a slice of atoms is calculated from the fit of Eq. (2) as where k B is the Boltzman constant. This slightly overestimates the temperature, because it does not take into account that the atoms are 'anti-trapped', i.e. it neglects the contribution to the ∆x t that originates during the lasing phase from the negative curvature of the potential of Eq. (1).

Beam Divergence
One can determine the divergence Θ of an atom laser beam either in real space or in momentum space. In real space one take the arctangent of the slope of a straight line fit to the size of the atom laser beam (∆x c ) versus the position x (Fig. 8a). In momentum space we take the arctangent of the calculated velocities for each slice separtely (Fig. 8b). The latter method requires knowledge of the initial size of the beams, which we estimate as the Thomas-Fermi radius of a BEC in the non-dressed trap containing all the atoms detected in the atom laser. The velocities are calculated from the position in the image on the basis of the gravitational and magnetic acceleration.
The results are not limited by the imaging resolution is 5 µm nor by the pixel size of 43 µm. The atoms in image slices from different positions have been outcoupled at different lasing times and might therefore have had different initial sizes. A more thorough analysis of the divergence would therefore have to follow the expansion of a single image slice.
For the 'pure' laser ( Fig. 8) the divergence in found a divergence of 10 ± 3 mrad in real and 13±6 mrad in momentum space. For the 'large flux' laser we found a divergence of 22±1 mrad in real space (Fig. 5b) and 13±2 mrad in momentum space (for the region of sufficient signal to noise, 0.7-13 mm).

Numerics
We consider the dynamics of the Gross-Pitaevskii equation with a dressed potential In order to reduce the computational complexity of the problem, we consider only the two dimensional problem. We introduce the dimensionless quantities τ = t/t 0 , ξ = y/y 0 , ζ = z/y 0 , and ψ = Aφ and then set the kinetic term coefficient to 1/2 and the nonlinear coefficient to unity by requiring that y 0 = h t 0 /M and A = M/(4πht 0 a s ). The resulting normalised Gross-Pitaevskii equation has the form with a potential and we have defined κ = Ω rf t 0 , α y = t 0 µ B αy 0 /h, F = B 0 t 0 µ B /h, α z = t 0 µ B βy 2 0 /(2h), G = gM y 0 t 0 /h, ∆(τ ) = 2πω rf (t)(τ t 0 )t 0 .
In Fig. 9, we see the dynamics of an atom laser BEC for different evolution times. We used a forth order split step Fourier method to computationally solve Eqs. (9)-(10). The scheme involves periodic boundary conditions, and thus, in order to "absorb" the wave close to the boundaries we introduced a boundary layer with linear loss. In Fig. 9 we note that there are only very small differences in the density profiles for τ = 500, 600, 700 and thus the dynamics are adiabatic and quasi-stationary. The small changes in the dynamics observed in the figure are attributed to small changes of the potential with time.

supplementary material
(i) A grayscale version of Fig. 1. (ii) The slices and fits used in the analysis of Fig. 1c, i.e. the atom beam containing both a thermal beam and an atom laser beam.
(iii) A video of the numeric simulation of an atom laser.